Next, we re-obtain Theorem 1 of [1]}.
Firstly, in (REF ),
we set \(\theta _{\textup {tsk}}=\theta _{\textup {env}}=1\) , and \(\mathrm {F}^{\textup {Task}} \left(a,b\right)=\mathrm {M}kl(a,b)\)  and \(\mathrm {F}^{\textup {Env}} \left(a,b\right)=2(-1)(a-b)^2\) .
These choices lead to \(k_e=1\)  \(\mathrm {G}_{\textup {Env}} \left(c\right)=\sqrt{\frac{c}{2(-1)}}\) , and
using further relaxation,
from \(\mathrm {M}kl(a,b)\le c_{\textup {tsk}}\) , it can be proved that
\(a\le b/(1-0.5 \lambda )+c_{\textup {tsk}}/(\mathrm {M}.\lambda (1-0.5\lambda ))\)  for \(\lambda \in (0,2)\) 
[2]}. Thus, we have \(k_t\) 
and \(\mathrm {G}_{\textup {Task}} \left(c\right)= c/(\mathrm {M}\lambda (1-0.5\lambda ))\) . It remains to obtain the log-terms of
\( \textup {B}_\textup {Task}\)  and
\( \textup {B}_\textup {Env}\)  given by (REF ) and (REF ).
Again, assuming the prior in independent of the data, by interchanging the order of expectations over \(\mathrm {P}_{T_{1:}}\mathrm {P}_{ \mathbf {Z}_{1:}^{\mathrm {M}}|T_{1:}}\)  and priors, we find (), and
\(\mathbb {E}_{ \mathcal {P}\mathrm {P}} \mathbb {E}_{\mathrm {P}_{T_{1:}}}\mathbb {E}_{\mathrm {P}_{\mathbf {Z}_{1:}^{\mathrm {M}}|T_{1:}}} e^{\mathrm {M}kl\left(\mathrm {L}_{\mathrm {P}_{Z|T_i}}(W),\mathrm {L}_{\mathbf {Z}_i^{\mathrm {M}}}(W)\right)}\le 2\sqrt{\mathrm {M}},\) 
